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The Origin of Euclid’s Axioms*
Published online by Cambridge University Press: 03 November 2016
Extract
Early in the classical period of Greek mathematics there arose a crisis over the meaning of “proportion”, or alternatively of “area”. This crisis, which was resolved by Eudoxus, the pupil of Plato, about the middle of the fourth century B.C., led to the introduction of axioms, or postulates, as “self-evident truths”, necessary for the proof of theorems, equally true but presumably less self-evident. Among these axioms was included the famous Fifth Postulate of Euclid, equivalent to saying (see below for its rather complicated wording) that “through a given point (the point T in Fig. 1) there is at most one (straight) line parallel to a given line (the line MK)”. But this postulate had all the earmarks of a theorem; it looks true yet not altogether self-evident, and its converse, to the effect that there is at least one such line, is an easily provable theorem.
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- Copyright © Mathematical Association 1961
Footnotes
Under the title “Mathematics” this article will form a chapter in the volume “The Great Transition in Ancient Greece: Studies of the Fourth Century B.C.” in preparation by the Department of Classics of Brown University, Providence, U.S.A.
References
* Under the title “Mathematics” this article will form a chapter in the volume “The Great Transition in Ancient Greece: Studies of the Fourth Century B.C.” in preparation by the Department of Classics of Brown University, Providence, U.S.A.
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